\section{Outline on weather variables}\label{sec:weather_variables}
\subsection{Geopotential Height}
The primary part of our study focuses on the establishment of the so-called \textit{weather regimes}: meteorological configurations at a regional/sub-continental level, defining a proper atmospheric state, and highly influencing climate phenomena such as precipitations, winds, water levels, temperatures etc.

\begin{figure}[!htbp]
    \centering
    \begin{subfigure}{0.3\textwidth}
            \centering
            \includegraphics[width=\linewidth]{imgs/example_z.png}
            \caption{Observation of geo-potential height at 500hPa (Z500)}
            \label{fig:example_z}
    \end{subfigure}%
    \hfill
    \begin{subfigure}{0.3\textwidth}
            \centering
            \includegraphics[width=\linewidth]{imgs/example_normal.png}
            \caption{30-years ``flat" normal of geo-potential height at 500hPa (Z500).}
            \label{fig:example_normal}
    \end{subfigure}%
    \hfill
    \begin{subfigure}{0.3\textwidth}
            \centering
            \includegraphics[width=\linewidth]{imgs/example_anomaly.png}
            \caption{Anomaly of geo-potential height at 500hPa (Z500)}
            \label{fig:example_anomaly}
    \end{subfigure}%
    \caption{Geo-potential maps of daily observation, normal and anomaly}
\end{figure}

They are usually addressed considering the \textit{geopotential height} (gpH) or Sea Level Pressure (SLP) variable: the altitude at which a constant atmospheric pressure level is measured (i.e. the altitude of an iso-pressure atmospheric level curve). Normally, the reference pressure value is Z500, corresponding to 500 hPa (hecto-Pascal), and taking into account those physical phenomena occurring in the mid/low Stratosphere, thus majorly driving climate events at the surface level. However, across the studies other pressure levels have shown to be suitable to accomplish the task \parencite{WR700hPa, Vautard1988, meteofrance, article}.\\
Instead of considering the raw gpH observation, the so-called \textit{anomaly} (Figure \ref{fig:example_anomaly}) is examined. It is a physical quantity addressing the variation with respect to the \textit{normal} behavior (Figure \ref{fig:example_normal}).\\
We define the normal as corresponding to an average involving (typically) the last 30 years of observations of the considered period of the year, thus resembling the mean expected behavior. It is a position-wise quantity, that, in the case of the geopotential height, shows a clear trend: it increases around the tropics (typically characterized by higher-pressure regimes), and decreases moving towards extreme latitudes.\\
Evaluating it often requires some processing, in order to eliminate any sort of seasonality that could be present when considering periods spanning across different seasons. Further, it has proved to highly influence the sensitivity of algorithms in the modeling pipeline, especially when leading to large anomaly values. For this reason, in our project we consider two types of normal: a ``flat'' normal, independent of the time, and a more ``dynamic'' normal, varying at different temporal granularity levels (month, week, and day).


Pressure zones are generally divided between low and high pressure zones, consequently corresponding to low and high values of geopotential height. In a 3D coordinate systems, they can be mathematically conceived as local minima/maxima on the isobaric plane. Under a meteorological point of view, low pressure zones are usually drivers of precipitations and unstable climatic conditions. Conversely, high pressure areas tend to be drier and with more stable setups. The geographical zone between an high pressure and a low pressure clusters experiences instead a general windy climate, the more accentuated, the closer the two clusters are. Finally, according to the relative positions of the two pressure masses, wind currents take either clockwise or anti-clockwise directions.\\
Thereby, weather regimes are recurrent, and quasi-persistent spatial distributions of these pressure masses, presenting a well-defined relative location of positive and negative pressure zones. They vary across geographical areas \parencite{LargeScaleWeatherRegimesandLocalClimateovertheWesternUnitedStates, asia}, and between different seasons, being more prominent during winter and summer \parencite{summer_regimes_cassou}. 

\begin{figure}[!htbp]
    \centering
    \includegraphics[width = .6\textwidth]{imgs/Regimes.jpeg}
    \caption{Visualization of the four weather regimes \parencite{weather_regimes} in the North-Atlantic and European zone. From left to right, up to down: NAO+, presenting high pressure in the mid-latitude level, and low pressure over the sub-arctic zone; Scandinavian Blocking (SB), characterized by high pressure over the Scandinavian peninsula; Atlantic Ridge (AR) configuration presenting higher pressure over the Atlantic Ocean region; NAO-, reporting general low pressure over Europe and high pressure in the sub-arctic region}
    \label{fig:weather_regimes}
\end{figure}

Our project is mainly centered on the North-Atlantic and European zone during the winter season. In such context, the four theoretical regimes (Figure \ref{fig:weather_regimes}) \parencite{strauss, grams, Falkena2020RevisitingTI, summer_regimes_cassou, 2011ClDy...36...19C, van_der_Wiel_2019} are:
\begin{itemize}
    \item \textit{NAO positive} and \textit{NAO negative} (NAO+ and NAO-, respectively). North Atlantic Oscillations represent the two main weather configurations, respectively localizing high pressure (low pressure for NAO-) systems in the Mediterranean Europe/mid-Atlantic region, and low pressure (respectively, high pressure) in the northern Europe/Greenland.\\
    Under NAO+, winters are generally warmer over all Europe, with some precipitations localizing in the north-western zone of the continent. The Atlantic is instead more likely interested by storms.\\
    NAO- is characterized by colder winters in north-western Europe, and a wetter season in south-eastern Europe. Overall, there is less probability of winter storms.
    \item \textit{Scandinavian Blocking} (SB). As the name recalls, the high pressure interests the Scandinavian peninsula and the northern Europe, while low pressure is mostly present over the Greenland and northern Atlantic America.\\
    Under SB, winters are typically dry and cold in the wester and central zones of Europe.
    \item \textit{Atlantic Ridge} (AR). Again, as the name suggests, high pressure is over the entire Atlantic Ocean region, while low pressure is concentrated over the Scandinavia and, more limitedly, on the Europe.\\
    Under AR, temperatures are slightly below normal over Europe, and typically dry in the western part.
\end{itemize}
NAOs tend to be the pre-dominant regimes during winter, accounting for most of the variability. Due to this, we often resort to a custom indicator called \textit{NAO Index} \parencite{naoindex}, whose positive values are associated to NAO+ persistances, while negative scores are more resembling NAO-. Further, its amplitude is often used to track extreme events, whether magnitudes are extreme.